Global Minima Problem

This is an example of Simulated Annealing for finding global minima of a function value.

Problem Description

Find the value of \(x_1\) and \(x_2\) to make the following function return closest number to zero where \(-1 \leq x_1, x_2 \leq 1\).

\[ f(x_1, x_2) = 0.2 \times {x_1}^2 + {x_2}^2 - 0.1 \times \cos(6 \times 3.1415 \times x_1) - 0.1 \times \cos(6 \times 3.1415 \times x_2) \]

Implementation

First, we need several library before implements the SA (simulated annealing) algorithm.

library(pracma)
library(plotly)

Then, we define the function in R. Also, we will choose a random starting point and creating a Mesh Grid for visualization.

f <- (function(x1, x2)
  0.2 * (x1 * x1) + (x2 * x2) - 0.1 * cos(6 * 3.1415 * x1) - 0.1 * cos(6 * 3.1415 * x2)
)
x_start <- c(0.7, -0.2)
mesh_grid <- meshgrid(seq(-1.0, 1.0, by = 0.01), seq(-1.0, 1.0, by = 0.01))
mesh_x1 <- mesh_grid$X
mesh_x2 <- mesh_grid$Y
f_mesh <- mesh_grid
for (i in range(length(mesh_x1[[1]])))
  for (j in range(length(mesh_x2[[2]])))
    f_mesh[[i]][[j]] <- f(mesh_x1[[i]][[j]], mesh_x2[[i]][[j]])
plot_ly(z = f_mesh$Y * f_mesh$X, type = "contour", contours = list(coloring = "heatmap", showlabels = TRUE))